Contour-time approach to the Bose-Hubbard model in the strong coupling regime

We study correlations in space and time after a quantum quench in the Bose Hubbard model. We obtain a two particle irreducible effective action within the contour-time formalism that allows us to study both equilibrium and out of equilibrium phenomena in the superfluid and Mott insulating phase regimes of parameter space. We derive equations of motion for both the superfluid order parameter and two-point correlation functions. To assess the accuracy of this formalism, we study the equilibrium solution of the equations of motion and compare our results to existing strong coupling methods as well as exact methods where possible. We discuss possible applications of this formalism to out of equilibrium scenarios.